In this project work will be done on self-conformal mappings of Riemann surfaces. In recent work the principal investigator has given a representation for arbitrary Riemann surfaces which provided simple proofs of old results together with new applications to self-conformal maps of surfaces which have a fixed point. In cases where the group of maps have several base points good progress has been made in describing the groups in terms of congruent polygons. Related to this work is on-going research measuring the number of regular mappings between a fixed (closed) Riemann surface to surfaces of lower genus. This finite bound depends only on the genus of the original surface. Similar questions regarding maps between multiply connected plane domains will be treated. The extremal metric method and Loewner's variational method of studying univalent functions have never been studied for connections. A careful analysis of this question will be undertaken. Aside from general interest, the outcome of this effort should provide a characterization of Loewner chains associated with quadratic differentials.
